Remanence

As the applied field is reduced to 0 after the sample has reached saturation the sample can still possess a remanent magnetisation, $M_{r}$. The magnitude of this remanent magnetisation is a product of the saturation magnetisation, the number and orientation of easy axes and the type of anisotropy symmetry. If the axis of anisotropy or magnetic easy axis is perfectly aligned with the field then $M_{r} \cong M_{s}$, and if perpendicular $M_{r} \cong 0$.

At saturation the angular distribution of domain magnetisations is closely aligned to $H$. As the field is removed they turn to the nearest easy magnetic axis. In a cubic crystal with a positive anisotropy constant, $K_{1}$, the easy directions are $\langle100\rangle$. At remanence the domain magnetisations will lie along one of the three $\langle100\rangle$ directions. The maximum deviation from $H$ occurs when $H$ is along the $\langle111\rangle$ axis, giving a cone of distribution of 55$^{\circ}$ around the axis.[10] Averaging the saturation magnetisation over this angle gives a remanent magnetisation of $0.832\,M_{s}$

In a system with uniaxial anisotropy with positive $K_{1}$ at remanence the magnetisation vectors cover a hemicircle in a two-dimensional system and a hemisphere in a three-dimensional system. These give $M_{r} \cong 0.637$ and $M_{r} = 0.5$ respectively.[9]

These situations are for ideal cases and can be modified greatly by further interactions and sample characteristics.